Axioms for generalized graphs, illustrated by a Cantor–Bernstein proposition

Abstract.

The notion of a graph type $\mathcal{T}$ is introduced by a collection of axioms. A graph of type $\mathcal{T}$ (or $\mathcal{T}$-graph) is defined as a set of edges, of which the structure is specified by $\mathcal{T}$. From this, general notions of subgraph and isomorphism of $\mathcal{T}$-graphs are derived. A Cantor-Bernstein (CB) result for $\mathcal{T}$-graphs is presented as an illustration of a general proof for different types of graphs. By definition, a relation $\mathcal{R}$ on $\mathcal{T}$-graphs satisfies the CB property if $A \mathcal{R} B$ and $B \mathcal{R} A$ imply that A and B are isomorphic. In general, the relation ‘isomorphic to a subgraph’ does not satisfy the CB property. However, requiring the subgraph to be disconnected from the remainder of the graph, a relation that satisfies the CB property is obtained. A similar result is shown for $\mathcal{T}$-graphs with multiple edges.